Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 498

Exercise M.7.5 (MATLAB file exercise_M_7_5.m ) Given a normal noise with
m ¼ 4 and s 2 ¼ 4.

1. If the noise is filtered with a low-pass filter H 1 ( z ), using the periodogrammethod,
estimate the PSD and plot the PSD at the filter output.
2. If the noise is filtered with a high-pass filter H 2 ( z ), using the periodogram
method, estimate the PSD and plot the PSD at the filter output.
Solution
1. The estimated PSDs of the input and output signals are shown in Fig. 7.42 . The
filter attenuates the high frequencies of the input signal.
2. The estimated PSDs of the input and output signals are shown in Fig. 7.43 .The
filter attenuates the low frequencies of the input signal including a DC component.

Exercise M.7.6 (MATLAB file exercise_M_7_6.m ) A normal noise with m ¼ 0
and s 2

¼ 16 is LP filtered and then multiplied with a sinusoidal signal of amplitude
A ¼ 1 and f ¼ 300 Hz.
Estimate and plot the PSDs at the filter output and at the product device output.

Solution The estimated PSDs are shown in Fig. 7.44 . Note that the PSD at the
output of the product device has replicas of the input spectrum around the sinusoi-
dal frequency f ¼ 300 Hz or o / p ¼ 300/500 ¼ 0.6.

Exercise M.7.7 (MATLAB file exercise_M_7_7.m ) Given two independent
low-pass and high-pass normal signals y 1 ( t ) and y 2 ( t ), respectively, both with
m ¼ 0 and s 1 2 ¼ 16, s 2 2 ¼ 1. A sinusoidal signal x ( t ) of amplitude A ¼ 1 and

f ¼ 250 Hz is added to the signal y 2 ( t ) yielding

LP filtering: I nput signal

10 5

10 0

10 -5

-1 -0.5 0 0.5 1

w / p

LP filtering: Output signal

10 5

10 0

10 -5

-1 -0.5 0 0.5 1

w / p

Fig. 7.42 LP filtering in Exercise M.7.5